Geometry both discrete and continuous at once, like information-Kempf

Geometry both discrete and continuous at once, like information--Kempf

It is possible for a geometry to be both discrete and continuous. We don't know if our universe's geometry is like that, but it could be. Video of a talk at Perimeter by Achim Kempf, describing this, was put online yesterday.

It refers to this paper published in Physical Review Lettershttp://arxiv.org/abs/0708.0062On Information Theory, Spectral Geometry and Quantum Gravity
Achim Kempf, Robert Martin
4 pages
(Submitted on 1 Aug 2007)
"We show that there exists a deep link between the two disciplines of information theory and spectral geometry. This allows us to obtain new results on a well known quantum gravity motivated natural ultraviolet cutoff which describes an upper bound on the spatial density of information. Concretely, we show that, together with an infrared cutoff, this natural ultraviolet cutoff beautifully reduces the path integral of quantum field theory on curved space to a finite number of ordinary integrations. We then show, in particular, that the subsequent removal of the infrared cutoff is safe."

and also to this paper Kempf recently posted on arxiv:http://arxiv.org/abs/0908.3061Information-theoretic natural ultraviolet cutoff for spacetime
Achim Kempf
4 pages
(Submitted on 21 Aug 2009)
"Fields in spacetime could be simultaneously discrete and continuous, in the same way that information can: it has been shown that the amplitudes, [tex]\phi(x_n)[/tex], that a field takes at a generic discrete set of points, [tex]x_n[/tex], can be sufficient to reconstruct the field [tex]\phi(x)[/tex] for all x, namely if there exists a certain type of natural ultraviolet (UV) cutoff in nature, and if the average spacing of the sample points is at the UV cutoff scale. Here, we generalize this information-theoretic framework to spacetimes themselves. We show that samples taken at a generic discrete set of points of a Euclidean-signature spacetime can allow one to reconstruct the shape of that spacetime everywhere, down to the cutoff scale. The resulting methods could be useful in various approaches to quantum gravity."

Jal says the PIRSA video is of the same talk that Kempf gave in Vancouver last month, at the EG4 conference.

Re: Geometry both discrete and continuous at once, like information--Kempf

There is a sense in which the main point is obvious, just the consequence of assuming a UV cutoff. So one possible take is a kind of smartpants "duh! that's trivial" reaction.
But if you watch the video you see he has brought considerably more to it. And the first 4 page paper was published in PRL for a reason. I think the second 4 page paper is also tailored for PRL. The video lecture is a good presentation. It was pitched for advanced undergrad students but there were interested senior people like Lee Smolin in the audience asking questions. Kempf took some care to make the ideas accessible to everybody, without diminishing their interest.

The trivial point about the cutoff, obvious once somebody points it out, is shown by the analogy with CD music, he mentioned a piece by Mozart.
Since the human ear only hears up to 20 kHz, the same sound can be in the form of both analog info or digitized discrete info. Two different mathematical models representing the same thing.

In an analogous way, geometry+matter might have two equally valid mathematical representations, one discrete and the other continuous. It might be meaningless to ask which is "right".

That much of the idea is kind of straightforward, the thing is once that is said what do you do with the idea, where do you go with it? I think Kempf does some interesting things with it.

Re: Geometry both discrete and continuous at once, like information--Kempf

Apart from the obvious need to justify the cutoff, which hopefully he does, the Nyquist theorem has a continuous background - the discrete sampling must be done infinitely precisely in continuous time in order to reconstruct the music.

The condensed matter models are discrete, but they all have a continuous background.

I don't think anyone would call those examples fundamental discreteness - Wen says it sometimes, but I don't buy it (yet?).

So I don't find the analogy very helpful, or if it is, he must be saying something trivial.

Jal, besides verifying the cutoff, I don't see the relevance of experiment right away, let me think about it.

Verifying the cutoff means verifying that you can't measure finer than a planck length. It is something that I think would be very hard to establish empirically, even if a very clever use of cosmological observation was made.

The cutoff on measurement is something people tend to ASSUME because it is reasonable, or they think it's reasonable. If you believe the HUP and that GR applies down close to Planck scale, then trying to measure a planck length creates such a big uncertainty of momentum, which by GR creates such a big uncertainty of curvature, that it backreacts on the length, so to speak, and makes the length measurement meaningless.

In other words if they are true down at P scale, then HUP and GR conspire together to foil your attempt to measure---so the cutoff on measurement is so to speak established by a thought experiment.

It is up to you how seriously you take this.

If you take it seriously then it means it is possible that the same mathematical model can be written down either as discrete or continuous..
In such a context it doesn't make sense to ask if Nature is discrete or continuous.

To make this more precise you may need to watch Kempf's talk (again), and not rely on anybody's paraphrase.

...
"We show that there exists a deep link between the two disciplines of information theory and spectral geometry. This allows us to obtain new results on a well known quantum gravity motivated natural ultraviolet cutoff which describes an upper bound on the spatial density of information. Concretely, we show that, together with an infrared cutoff, this natural ultraviolet cutoff beautifully reduces the path integral of quantum field theory on curved space to a finite number of ordinary integrations. We then show, in particular, that the subsequent removal of the infrared cutoff is safe."

I have something essential to tell you about cut-offs. You probably know that Pomeranchuk and Landau tried to build QED with classically smeared electrons (or charges); they obtained something finite and then they made the smearing size tend to zero (point-like electron). In this limit the theory recovers its UV divergences, and that is why L.D. Landau concluded "Hamiltonian is dead". In fact, in order to obtain finite results with a natural cut-off it is sufficient to have a quantum mechanical rather than classical smearing.

But where to take a quantum mechanical smearing for an electron? From QED, of course: the quantized EMF makes the electron move and this gives the most natural QM smearing in the theory. A model of such a QED (Novel QED) is given in my publications. My approach describes naturally both the Lamb shift and the soft radiation and no IR divergences appear.

I believe that the other theories should be constructed similarly to the Novel QED.

Re.: Cut off
Until we find a way of finding experimental evidence ... its my beliefs vs your beliefs.

That is why it is essential to understand what is QM charge smearing, how it depends on the external field and how it is observed since long-long ago. For example, a point-like electron is the inclusive picture, average, not exact.

Re: Geometry both discrete and continuous at once, like information--Kempf

I for one, think the idea to look for some sort of physically justifiable "cutoffs" is interesting.

Just to throw in a reflection from my own perspective, one way I imagine how to sort of "verify" a cutoff, is that when you consider that this cutoff has implications for the computation of the standard model action, there could be (remains to find out though) that such a cutoff has implicatations for the actions at the planck scale.

In my personal inference view of physics, the constraints on actions seems to be the only reasonable feedback to such a conjecture.

The analogy of the human and mozart would be the the action of the human would be indifferent to wether the sound is coming from an analog tape, or a digital CD. If the human is truly indifferent, then that's what we want to know.

Similarly, if the action of a physical system is indifferent to a cutted version or the som imagined "full version", then that's all that counts. I personally associate the sampling truncating to the limed information capacity of the observer, and with the note that the observer is the system wich must encode/implement the actual sampling machinery.

I think something like that is what we are looking for. But the exact formulation seems hard to nail.

But it seems to me that one thing seems fairly clear, that this cutoff is not something fundamental to the object under study, it's somehow a relation between the observer and the observed, since the observers complexity must somehow enter the picture. That's my opinon at least.

Therefore I think also this cutoff of spacetime and similar things, are not a property of spacetime alone, I think one has to see it from the point of view of a physical systems. It's probably partly in the eye of the beholder. This is why I think one has to have both matter and space to make sense of this. If space is somehow encoded in the microstructure of matter, then a link could be the discreteness of matter, implies that this same matters action is indifferent to a certain excess of complexity in it's own environment. Perhaps what's left in the action system is a kind of discrete distinguishable "spacetime".